Equation Balance
2025-07-25
A simple system.
y_{1}^{*} = X_{1} \beta_{1} + \alpha_{1} y_{2}^{*} + \epsilon_{1} y_{2}^{*} = X_{2} \beta_{2} + \alpha_{2} y_{1}^{*} + \epsilon_{2}
What are these rank and order conditions?
In a system of G equations any particular equation is identified if and only if it is possible to construct at least one non-zero determinant of the order (G-1) from the coefficients excluded from that particular equation but contained in other equations of the model.
A sufficient condition for the identification of a relationship is that the rank of the matrix of parameters of all the excluded variables (endogenous and pre-determined) from that equation be equal to (G-1).
This is an application of the rank of a matrix – the number of linearly independent columns.
In order to check the rank condition for the first equation we have to proceed as follows: Delete the first row and collect the columns for those variables of the first equation that were marked with zero. For equation 1, y and X2 was marked with zero, and if we collect those two columns we receive:
If this matrix contains less than M-1 rows or columns where all elements are zero, equation 1 will not be identified. m refers to the number of equation just as in the order condition, which means that M-1=2. Since we have two rows and two columns and none of them contains only zeros we conclude that equation 1 is identified. Citation:
Let: G = total number of equations (total number of endogenous variables. K= total number of variables in the model (endogenous and pre-determined). M = number of variables, endogenous and pre-determined, in a particular equation.
K-M \geq G-1
The order condition is a necessary condition for identification but it is not sufficient.
To acquire estimates of the system, each equation must be at least identified. To confirm this, one must at least utilize the rank condition from above. Why is identification important? It means that there is at least some unique information applied to each endogenous element.
We are most often concerned with non-stationarity in the mean.
There is an entire suite of models, particularly used in finance applications with high frequency data focused on variance. These are known as AutoRegressive Conditional Heteroscedasticity Models or ARCH models.
A simple univariate model:
y_t = a_0 + a_1 y_{t-1} + e_t
where
e_t = v_t h_{t}^{\frac{1}{2}} with v_t as white noise. Thus, the conditional variance of the series is given by
h_t = \alpha_{0} + \alpha_{1} e^{2}_{t-1} ## Finding ARCH
Fractional integration methods. Structural breaks: see Jong Hee Park’s work. Unit roots in the presence of breaks…
Their definition
Can apply to both the theoretical and the empirical model.
Andy Phillips has a recent paper on inference in dynamic settings. It is in the box for day 5. The paper is:
How to avoid incorrect inferences (while gaining correct ones) in dynamic models
ESSSSDA25-2W: Heterogeneity and Dynamics